They first show that a valid belief update should minimise an overall loss \(L\)  of the form \(L=E_\nu (l)+D(\nu ,\pi _0)\) , where the second term is a measure \(D(\nu ,\pi _0)\)  of divergence between prior and \(\nu \) . For our purposes this actually defines what we mean by a “valid" belief update. [1]} and [2]} show that if the loss \(l\)  is additive, and the belief update \(\psi \)  determined by (REF ) is required to be order-coherent whatever the prior, parameter space, loss and data it is updating, if \(L\)  has a unique minimum and \(D=D_g\)  is a \(g\) -divergence (see Appendix REF ) then \(D_g\)  must be the KL-divergence and so a valid coherent belief update must minimise (REF ).
